Solve each system of equations using matrices (row operations). If the system has no solution, s

step 1 Okay, so let's look at this problem. First step, okay, turn it into a matrix. Negative 1, 1, 1,

Solve each system of equations using matrices (row...

Key Concepts

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by systematically applying elementary row operations to transform the augmented matrix into an upper triangular or row echelon form. This process simplifies the system, allowing for easier determination of the solution through back substitution.

Elementary Row Operations

Elementary row operations include interchanging rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another row. These operations are used to simplify the augmented matrix to a form that makes it easier to directly read off the solutions of the system.

Inconsistency and Consistency in Systems

A system is consistent if there is at least one set of values for the variables that satisfies all the equations simultaneously; it is inconsistent if no such set exists. When performing row operations, a row that results in all zeros for the coefficients but has a non-zero constant indicates the system is inconsistent.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share common variables. The solution to the system is found by identifying values for the variables that satisfy all the equations simultaneously. Such systems can have a single solution, infinitely many solutions, or no solution at all.

Augmented Matrix

An augmented matrix is a structured representation of a system of linear equations. It includes both the coefficients of the variables and the constants from the right-hand side of the equations in a single matrix format. This representation is especially useful in applying systematic techniques like row operations to solve the system.

Transcript

00:01 Okay, so let's look at this problem.

00:04 First step, okay, turn it into a matrix.

00:09 Negative 1, 1, 1, and negative 1, 2, negative 3, negative 2, negative 7, negative 2, negative 7.

00:21 Negative 1, 94, 0.

00:26 Okay, so i label this, row 1, row 2, and row 3.

00:31 So the first role operation want to do is we want to use the third row plus three times the first row and we assign this new value to the third row.

00:54 So after this role operation, what we have here is, okay, let's change another color maybe.

01:04 Okay, let's change it to gray.

01:06 Okay, so what we have here is 91, okay, we'll just, okay, we'll just, just copy down the first two rows, because they are unchanged, right? and the third row is negative 0, 1, negative 4, and negative 3.

01:51 Okay, so the next role operation we want to do is we want to turn this to 0.

01:59 So what do we do is we use that second rule.

02:03 Okay, let's label here, row 1, row 2, row 3.

02:06 We use row 2 minus row 1 times 2, sorry, just minus row 1 and we assign this new value to row 2.

02:17 Okay, so now after the operation we have negative 1, 1, 1, 1.

02:23 Okay, copy the first row.

02:26 Okay, the second row is just 0 -1, negative -4, and negative 3.

02:33 And then the third row is 0 -1, 95 -194, negative 3.

02:38 Okay, you can see that these two rows are the, actually the, the same, okay.

02:44 So we can use the third row to subtract the second row, okay? so we assign this value to the third rule.

02:55 Okay, this is row one, row two, and row three.

03:02 We use row three, subtract row two, and then we assign this to row three.

03:08 Okay, so we have zero zero zero, 0, okay.

03:28 And the next step we want to do is we want to use the first role, use the first row and subtract the third rule and we assign this new value to, sorry, the second row, subtract the second row, and then we assign this new value to the first row...

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